Doubling rules [cellular automata]

Rule (a) is

{{0, 2, _}  5, {5, 3, _}  5, {5, _, _}  1, {_, 5, _}  1, {_, 2, _}  3, {_, 3, 2}  2, {_, 1, 2}  4, {_, 4, _}  3, {4, 3, _}  4, {4, 0, _}  2, {_, x_, _}  x}

and takes 2n² + n steps to yield Table[1, {2n}] given input Append[Table[1, {n - 1}], 2]. Rule (b) is

{{_, 2, _}  3, {_, 1, 2}  2, {3, 0, _}  1, {3, _, _}  3, {_, 3, _}  1, {_, x_, _}  x}

and takes 3n steps. Rule (c) is k = 3, r = 1 rule 5407067979 and takes 3n - 1 steps.